Question 827529
Try assigning variables to everything.


x = length
y = width
A = 43.56 cm^2
B = 63.84 cm^2
c = 1.20 cm length increment to obtain B area


Form equations.


{{{xy=A}}}, and {{{(x+c)(y+c)=B}}}.
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{{{y+c=B/(x+c)}}}
{{{y=B/(x+c)-c}}}
{{{y=B/(x+c)-c(x+c)/(x+c)}}}
{{{y=(B-cx-c^2)/(x+c)}}}
'
Original area equation, A,
{{{xy=A}}}
{{{x((B-cx-c^2)/(x+c))=A}}}
{{{x(B-cx-c^2)=A(x+c)}}}
{{{Bx-cx^2-c^2*x=Ax+Ac}}}
{{{-Bx+cx^2+c^2*x=-Ax-Ac}}}
{{{cx^2+c^2*x+Ax-Bx=-Ac}}}
{{{cx^2+c^2*x+Ax-Bx+Ac=0}}}
{{{highlight(cx^2+(c^2+A-B)x+Ac=0)}}}


The time required to enter all of that as text and put in the rendering brackets  is very long.  Solve the quadratic equation for x and use the given values according to you best comfort to get a value for x; and use it to find y.
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Note that you might want to compute the coefficients of the constant term, of x^2, and of x; before using the general solution to quadratic equation.
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I did NOT assume any convenience with the given numeric values.